NCERT solutions for Class 12 Physics Chapter 3 Current Electricity

3.1 The storage battery of a car has an emf of 12 V. If the internal resistance of the battery is 0.4 $\mathrm{\Omega }$ , what is the maximum current that can be drawn from the battery?

Answer :

Given, the emf of battery, E = 12 V

Internal resistance of battery, r = 0.4 Ohm

Let I be the maximum current drawn from the battery.

We know, according to Ohm's law

E = Ir

I = E/r = 12/0.4 =30 A

Hence the maximum current drawn from the battery is 30 A.

3.2 A battery of emf 10 V and internal resistance 3 Ω is connected to a resistor. If the current in the circuit is 0.5 A, what is the resistance of the resistor? What is the terminal voltage of the battery when the circuit is closed?

Answer:

Given, The emf of the battery, E = 10 V

The internal resistance of the battery, r = 3 Ohm

Current in the circuit, I = 0.5 A

Let R be the resistance of the resistor.

Therefore, according to Ohm's law:

E = IR' = I(R + r)

10 = 0.5(R + 3)

R = 1 Ohm

Also,

V = IR (Across the resistor)

= 0.5 x 17 = 8.5 V

Hence, the terminal voltage across the resistor = 8.5 V

3.3 At room temperature (27.0 °C) the resistance of a heating element is 100 Ω. What is the temperature of the element if the resistance is found to be 117 Ω, given that the temperature coefficient of the material of the resistor is 1.7 x 10^{-4 o}C^-1

Answer:

Given,

temperature coefficient of filament,

α = 1.7 x 10^{-4 o}C^-1

$T_1={27}^{\circ }C$ ; $R_1=100\mathrm{\Omega }$

Let T₂ be the temperature of element, R₂=117 Ω

(Positive alpha means that the resistance increases with temperature. Hence we can deduce that T₂ will be greater than T₁ )

We know,

$R_2=R_1[1+\alpha \mathrm{\Delta }T]$

$⟹$ $117=100[1+(1.70\times {10}^{-4})(T_2-27)]$

$$\begin{gathered} \Rightarrow T_2-27=\frac{117-100}{1.7\times {10}^{-4}} \\ \Rightarrow T_2-27=1000 \\ \Rightarrow T_2={1027}^{\circ }C \end{gathered}$$

Hence, the temperature of the element is 1027 °C.

3.4 A negligibly small current is passed through a wire of length 15 m and uniform cross-section $6.0\times {10}^{-7}{m}^2$ , and its resistance is measured to be 5.0 Ω. What is the resistivity of the material at the temperature of the experiment?

Answer:

$R=\rho l/A$ , where $\rho$ is the resistivity of the material

$⟹\rho =RA/l=5\times 6\times {10}^{-7}/15$

$⟹\rho =2\times {10}^{-7}$

Hence, the resistivity of the material of wire is $\rho =2\times {10}^{-7}m$

3.5 A silver wire has a resistance of 2.1 Ω at 27.5 °C, and a resistance of 2.7 Ω at 100 °C. Determine the temperature coefficient of resistivity of silver.

Answer:

Given,

$T_1={27.5}^{\circ }C$ ; $R_1=2.1\mathrm{\Omega }$

$T_2={100}^{\circ }C$ ; $R_2=2.7\mathrm{\Omega }$

We know,

$R_2=R_1[1+\alpha \mathrm{\Delta }T]$

2.7 = 2.1[1 + $\alpha$ (100 - 27.5) ]

$\alpha$ = (2.7 - 2.1) / 2.1(100 -27.5)

$\alpha$ = 0.0039 ${}^{\circ }{C}^{-1}$

Hence, the temperature coefficient of silver wire is 0.0039 ${}^{\circ }{C}^{-1}$

3.6 A heating element using nichrome connected to a 230 V supply draws an initial current of 3.2 A which settles after a few seconds to a steady value of 2.8 A. What is the steady temperature of the heating element if the room temperature is 27.0 °C? Temperature coefficient of resistance of nichrome averaged over the temperature range involved is $1.70\times {10}^{-4}{}^{\circ }{C}^{-1}$.

Answer:

For the given voltage, the two values of current will correspond to two different values of resistance which will correspond to two different temperature.

V = 230 V

$I_1=3.2A$ and $I_2=2.8A$

Using Ohm's law:

$R_1=230/3.2=71.87\mathrm{\Omega }$

and

$R_2=230/2.8=82.14\mathrm{\Omega }$

$T_1={27}^{\circ }C$

Let $T_2$ be the steady temperature of the heating element.

We know,

$R_2=R_1[1+\alpha \mathrm{\Delta }T]$

That is

230/2.8 = 230/3.2[1 + ( $1.70\times {10}^{-4}$ ) ( T ₂ - 27) ]

T ₂ = (840.5 + 27) °C

Hence, steady temperature of the element is 867.5 °C.

3.7 Determine the current in each branch of the network shown in Fig. 3.30:

Answer:

The current in the circuit is distributed like

where I₁, I₂, and I₃ are the different currents through shown branches.

Now, applying KVL in Loop

$10-I10-I_{2}5-(I_2+I_3)10=0$

Also, we have $I=I_1+I_2$

So putting it in KVL equation

$10-(I_1+I_2)10-I_{2}5-(I_2+I_3)10=0$

$10-10I_1-10I_2-5I_2-10I_2-10I_3=0$

$10-10I_1-25I_2-10I_3=0$ .................................(1)

Now let's apply KVL in the loop involving I1 I2 AND I3

$5I_2-10I_1-5I_3=0$ .................................(2)

now, the third equation of KVL

$5I_3-5(I_1-I_3)+10(I_2+I_3)=0$

$-5I_1+10I_2+20I_3=0$ ..............................(3)

Now we have 3 equation and 3 variable, on solving we get

$I_1=\frac{4}{17}A$

$I_2=\frac{6}{17}A$

$I_3=\frac{-2}{17}A$

Now the total current

$I=I_1+I_2=\frac{4}{17}+\frac{6}{17}=\frac{10}{17}$

3.8 A storage battery of emf 8.0 V and internal resistance 0.5 Ω is being charged by a 120 V dc supply using a series resistor of 15.5 Ω. What is the terminal voltage of the battery during charging? What is the purpose of having a series resistor in the charging circuit?

Answer:

Emf of battery, E = 8 V

Internal resistance of battery, r = 0.5 Ω

Supply Voltage, V = 120 V

The resistance of the resistor, R = 15.5 Ω

Let V' be the effective voltage in the circuit.

Now, V' = V - E

V' = 120 - 8 = 112 V

Now, current flowing in the circuit is:

I = V' / (R + r)

$⟹I=\frac{112}{15.5+0.5}=7A$

Now, using Ohm ’s Law:

Voltage across resistor R is v = IR

V = 7 x 15.5 = 108.5 V

Now, the voltage supplied, V = Terminal voltage of battery + V

$\therefore$ Terminal voltage of battery = 120 -108.5 = 11.5 V

The purpose of having a series resistor is to limit the current drawn from the supply.

3.9 The number density of free electrons in a copper conductor estimated in Example 3.1 is $8.5\times {10}^{28}{m}^{-3}$ . How long does an electron take to drift from one end of a wire 3.0 m long to its other end? The area of cross-section of the wire is $2.0\times {10}^{-6}{m}^2$ and it is carrying a current of 3.0 A.

Answer:

We know,

$I=neA{v}_d$

v_d :drift Velocity = length of wire(l) / time taken to cover

$I=neA\frac{l}{t}$

by substituting the given values

$⟹$ t = 2.7 x 10⁴ s

Therefore, the time required by an electron to drift from one end of a wire to its other end is $2.7\times {10}^4$ s.

Additional Questions

1. The earth’s surface has a negative surface charge density of 10 ^-9 C m ^-2 . The potential difference of 400 kV between the top of the atmosphere and the surface results (due to the low conductivity of the lower atmosphere) in a current of only 1800 A cover the entire globe. If there were no mechanism of sustaining atmospheric electric field, how much time (roughly) would be required to neutralise the earth’s surface? (This never happens in practice because there is a mechanism to replenish electric charges, namely the continual thunderstorms and lightning in different parts of the globe). (Radius of earth = 6.37 × 10 6 m.)

Answer:

Given, the surface charge density of earth

$\rho$ = ${10}^{-9}C{m}^{-2}$

Current over the entire globe = 1800 A

Radius of earth, r = 6.37 x 10⁶ m

$\therefore$ The surface area of earth A = $4\pi r^2$

= $4\pi (6.37\times {10}^6{)}^2$ = $5.09\times {10}^{14}{m}^2$

Now, charge on the earth surface,

$q=\rho \times A$

Therefore,

$q=1005.09\times {10}^5\mathrm{C}$

Let the time taken to neutralize earth surface be t

$\therefore$ Current I = q / t

t = 282.78 s.

Therefore, time take to neutralize the Earth's surface is 282.78 s

2. Six lead-acid type of secondary cells each of emf 2.0 V and internal resistance 0.015 Ω are joined in series to provide a supply to a resistance of 8.5 Ω. What are the current drawn from the supply and its terminal voltage?

Answer:

Given,

There are 6 secondary cells.

Emf of each cell, E = 2 V (In series)

The internal resistance of each cell, r = 0.015 Ω (In series)

And the resistance of the resistor, R = 8.5 Ω

Let I be the current drawn in the circuit.

$I=\frac{nE}{R+nr}$

$⟹I=\frac{6(2)}{8.5+6(0.015)}=\frac{12}{8.59}$

$⟹$ I = 1.4 A

Hence current drawn from the supply is 1.4 A

Therefore, terminal voltage, V = IR = 1.4 x 8.5 = 11.9 V

3. A secondary cell after long use has an emf of 1.9 V and a large internal resistance of 380 Ω. What maximum current can be drawn from the cell? Could the cell drive the starting motor of a car?

Answer:

Given,

Emf , E = 1.9 V

Internal resistance, r =380 Ω

The maximum current that can drawn is I = E/r = 1.9/380 = 0.005 A

The motor requires a large value of current to start and hence this cell cannot be used for a motor of a car.

4. Two wires of equal length, one of aluminium and the other of copper have the same resistance. Which of the two wires is lighter? Hence explain why aluminium wires are preferred for overhead power cables. ( ${\rho }_{Al}=2.63\times {10}^{-8}\mathrm{\Omega }m$ . ${\rho }_{Cul}=1.73\times {10}^{-8}\mathrm{\Omega }m$ relative density of Al = 2.7, of Cu = 8.9.)

Answer:

We know,

R = Ρ l / A

The wires have the same resistance and also are of the same length.

Hence,

$\frac{{\rho }_{Al}}{A_{Al}}=\frac{{\rho }_{Cu}}{A_{Cu}}$

$⟹\frac{A_{Al}}{A_{Cu}}=\frac{{\rho }_{Al}}{{\rho }_{Cu}}=\frac{2.63}{1.73}$

Now, mass = Density x Volume = Density x Area x length

Taking the ratio of their masses for the same length

$⟹\frac{m_{Al}}{m_{Cu}}=\frac{d_{AL}{A}_{Al}}{d_{Cu}{A}_{Cu}}=\frac{2.7\times 2.63}{8.9\times 1.73}<1$

Hence, $m_{Al}<m_{Cu}$

Therefore, for the same resistance and length, the aluminium wire is lighter.

Since aluminium wire is lighter, it is used as power cables.

5. What conclusion can you draw from the following observations on a resistor made of alloy manganin?

Answer:

The ratio of Voltage to current for the various values comes out to be nearly constant which is around 19.7.

Hence the resistor made of alloy manganin follows Ohm's law.

Answer the following questions:

6. A steady current flows in a metallic conductor of non-uniform cross-section. Which of these quantities is constant along the conductor: current, current density, electric field, drift speed?

Answer:

The current flowing through the conductor is constant for a steady current flow.

Also, current density, electric field, and drift speed are inversely proportional to the area of cross-section. Hence, not constant.

Answer the following questions:

7. Is Ohm’s law universally applicable for all conducting elements? If not, give examples of elements which do not obey Ohm’s law.

Answer:

No. Ohm’s law is not universally applicable for all conducting elements.

A semiconductor diode is such an example.

Answer the following questions

8. A low voltage supply from which one needs high currents must have very low internal resistance. Why?

Answer:

Ohm's law states that: V = I x R

Hence for a low voltage V, resistance R must be very low for a high value of current.

Answer the following questions:

9. A high tension (HT) supply of, say, 6 kV must have a very large internal resistance. Why?

Answer:

A very high internal resistance is required for a high tension supply to limit the current drawn for safety purposes.

10. Choose the correct alternative:

(a) Alloys of metals usually have (greater/less) resistivity than that of their constituent metals.
(b) Alloys usually have much (lower/higher) temperature coefficients of resistance than pure metals.
(c) The resistivity of the alloy manganin is nearly independent of/ increases rapidly with increase of temperature.
(d) The resistivity of a typical insulator (e.g., amber) is greater than that of a metal by a factor of the order of ( ${10}^{22}/{10}^{23}$ ).

Answer:

(a) Alloys of metals usually have greater resistivity than that of their constituent metals.
(b) Alloys usually have much lower temperature coefficients of resistance than pure metals.
(c) The resistivity of the alloy manganin is nearly independent of temperature.
(d) The resistivity of a typical insulator (e.g., amber) is greater than that of a metal by a factor of the order of 10²² .

11. (a)

(i) Given n resistors each of resistance R , how will you combine them to get the (i) maximum effective resistance?

Answer:

To get maximum effective resistance, combine them in series. The effective resistance will be nR.

11. (b)

(ii) Given n resistors each of resistance R, how will you combine them to get the (ii) minimum effective resistance?

Answer:

To get minimum effective resistance, combine them in parallel. The effective resistance will be R/n.

11.(c)

(iii) What is the ratio of the maximum to minimum resistance?

Answer:

The ratio is nR/(R/n) = $n^2$

12.(a)

(i) Given the resistances of 1 Ω, 2 Ω, 3 Ω, how will be combine them
to get an equivalent resistance of (i) $\frac{11}{3}\mathrm{\Omega }$

Answer:

We have, equivalent resistance = 11/3

Let's break this algebraically so that we can represent it in terms of 1, 2 and 3

$\frac{11}{3}=\frac{9+2}{3}=3+\frac{2}{3}=3+\frac{1\ast 2}{1+2}$

this expression is expressed in terms of 1, 2 and 3. and hence we can make a circuit which consist only of 1 ohm, 2 ohms and 3 ohms and whose equivalent resistance is 11/3. that is :

12. (b)

(ii) Given the resistances of 1 Ω, 2 Ω, 3 Ω, how will be combine the to get an equivalent resistance of (ii) $\frac{11}{5}\mathrm{\Omega }$

Answer:

Connect 2 Ω and 3 Ω resistor in parallel and 1 Ω resistor in series to it

Equivalent Resistance R = {1/(1/2 + 1/3)} + 1 = 6/5 + 1

R = 11/5 Ω

12.(c)

(iii) Given the resistances of 1 Ω, 2 Ω, 3 Ω, how will be combine them to get an equivalent resistance of (iii) 6 Ω

Answer:

1 Ω+2 Ω+ 3 Ω= 6 Ω, so we will combine the resistance in series.

12.(d)

(iv) Given the resistances of 1 Ω, 2 Ω, 3 Ω, how will be combine them
to get an equivalent resistance of (iv) $\frac{6}{11}\mathrm{\Omega }$

Answer:

Connect all three resistors in parallel.

Equivalent resistance is R = 1/(1/1 + 1/2 + 1/3) = (1x 2 x 3)/(6 + 3 + 2)

R = 6/11 Ω

13.

(a) Determine the equivalent resistance of networks shown in
Fig. 3.31.

Answer:

It can be seen that in every small loop resistor 1 ohm is in series with another 1 ohm resistor and two 2 ohms are also in series and we have 4 loops,

so equivalent resistance of one loop is equal to the parallel combination of 2 ohms and 4 ohm that is

$Equivalent{R}_{loop}=\frac{2\ast 4}{2+4}=\frac{8}{6}=\frac{4}{3}$

now we have 4 such loops in series so,

$TotalEquivalent{R}_{loop}=\frac{4}{3}+\frac{4}{3}+\frac{4}{3}+\frac{4}{3}=\frac{16}{3}$

Hence equivalent resistance of the circuit is 16/3 ohm.

13.

(b) Determine the equivalent resistance of networks shown in Fig. 3.31.

Answer:

It can be seen that all 5 resistors are in series, so

Equivalent Resistance = R + R + R + R + R = 5R

Hence equivalent resistance is 5R.

14. Determine the current drawn from a 12V supply with internal resistance 0.5Ω by the infinite network shown in Fig. 3.32. Each resistor has 1Ω resistance.

Answer:

First, let us find the equivalent of the infinite network,

let equivalent resistance = R'

Here from the figure, We can consider the box as a resistance of R'

Now, we can write,

equivalent resistance = R'

' =[( R')Parallel with (1)] + 1 + 1

$\frac{R^{\prime }\ast 1}{R^{\prime }+1}+2=R^{\prime }$

$R^{\prime }+2R^{\prime }+2=R^{\prime 2}+R^{\prime }$

$R^{\prime 2}-2R^{\prime }-2=0$

$R^{\prime }=1+\sqrt{3},or1-\sqrt{3}$

Since resistance can never be negative we accept

$R^{\prime }=1+\sqrt{3}$

, We have calculated the equivalent resistance of infinite network,

Now

Total equivalent resistance = internal resistance of battery+ equivalent resistance of the infinite network

= 0.5+1+1.73

=3.23 ohm

$V=IR$

$I=\frac{V}{R}=\frac{12}{3.23}=3.72A$

Hence current drawn from the 12V battery is 3.72 Ampere.

15. Figure 3.33 shows a potentiometer with a cell of 2.0 V and internal resistance 0.40 Ω maintaining a potential drop across the resistor wire AB. A standard cell which maintains a constant emf of 1.02 V (for very moderate currents up to a few mA) gives a balance point at 67.3 cm length of the wire. To ensure very low currents drawn from the standard cell, a very high resistance of 600 kΩ is put in series with it, which is shorted close to the balance point. The standard cell is then replaced by a cell of unknown emf ε and the balance point found similarly, turns out to be at 82.3 cm length of the wire.

(a) What is the value of $ϵ$

Answer:

Given

Maintained constant emf of standard cell = 1.02V, balanced point of this cell = 67.3cm

Now when the standard cell is replaced by another cell with emf = $\epsilon$, balanced point for this cell = 82.3cm

Now as we know the relation

$\frac{\epsilon }{l}=\frac{E}{L}$

$\epsilon =\frac{E}{L}\ast l=\frac{1.02}{67.3}\ast 82.3=1.247V$

Hence, emf of another cell is 1.247V.

16. Figure 3.33 shows a potentiometer with a cell of 2.0 V and internal resistance 0.40 Ω maintaining a potential drop across the resistor wire AB. A standard cell which maintains a constant emf of 1.02 V (for very moderate currents upto a few mA) gives a balance point at 67.3 cm length of the wire. To ensure very low currents drawn from the standard cell, a very high resistance of 600 kΩ is put in series with it, which is shorted close to the balance point. The standard cell is then replaced by a cell of unknown emf ε and the balance point found similarly, turns out to be at 82.3 cm length of the wire.

(b) What purpose does the high resistance of 600 k $\mathrm{\Omega }$ have?

Answer:

If a sufficiently high current passes through galvanometer then it can get damaged. So we limit the current by adding a high resistance of 600 k $\mathrm{\Omega }$.

17. Figure 3.33 shows a potentiometer with a cell of 2.0 V and internal resistance 0.40 Ω maintaining a potential drop across the resistor wire AB. A standard cell which maintains a constant emf of 1.02 V (for very moderate currents upto a few mA) gives a balance point at 67.3 cm length of the wire. To ensure very low currents drawn from the standard cell, a very high resistance of 600 kΩ is put in series with it, which is shorted close to the balance point. The standard cell is then replaced by a cell of unknown emf ε and the balance point found similarly, turns out to be at 82.3 cm length of the wire.

(c) Is the balance point affected by this high resistance?

Answer:

No, the Balance point is not affected by high resistance. High resistance limits the current to galvanometer wire. The balance point is obtained by moving the joe key on the potentiometer wire and current through potentiometer wire is constant. The balance point is the point when the current through galvanometer becomes zero. The only duty of high resistance is to supply limited constant current to potentiometer wire.

18. Figure 3.33 shows a potentiometer with a cell of 2.0 V and internal resistance 0.40 Ω maintaining a potential drop across the resistor wire AB. A standard cell which maintains a constant emf of 1.02 V (for very moderate currents upto a few mA) gives a balance point at 67.3 cm length of the wire. To ensure very low currents drawn from the standard cell, a very high resistance of 600 kΩ is put in series with it, which is shorted close to the balance point. The standard cell is then replaced by a cell of unknown emf ε and the balance point found similarly, turns out to be at 82.3 cm length of the wire.

(d) Would the method work in the above situation if the driver cell of the potentiometer had an emf of 1.0V instead of 2.0V?

Answer:

No, the method would not have worked if the driver cell of the potentiometer had an emf of 1.0V instead of 2, because when emf of the driving point is less than the other cell, there won't be any balance point in the wire.

19. Figure 3.33 shows a potentiometer with a cell of 2.0 V and internal resistance 0.40 Ω maintaining a potential drop across the resistor wire AB. A standard cell which maintains a constant emf of 1.02 V (for very moderate currents upto a few mA) gives a balance point at 67.3 cm length of the wire. To ensure very low currents drawn from the standard cell, a very high resistance of 600 kΩ is put in series with it, which is shorted close to the balance point. The standard cell is then replaced by a cell of unknown emf ε and the balance point found similarly, turns out to be at 82.3 cm length of the wire.

(e) Would the circuit work well for determining an extremely small emf, say of the order of a few mV (such as the typical emf of a thermo-couple)? If not, how will you modify the circuit?

Answer:

No, the circuit would not work properly for very low order of Voltage because the balance points would be near point A and there will be more percentage error in measuring it. If we add series resistance with wire AB. It will increase the potential difference of wire AB which will lead to a decrease in percentage error.

20.

Figure 3.34 shows a 2.0 V potentiometer used for the determination of internal resistance of a 1.5 V cell. The balance point of the cell in open circuit is 76.3 cm. When a resistor of 9.5 Ω is used in the external circuit of the cell, the balance point shifts to 64.8 cm length of the potentiometer wire. Determine the internal resistance of the cell.

Answer:

Given,

the balance point of cell in open circuit = $l_1=76.3cm$

value of external resistance added = $R=9.5\mathrm{\Omega }$

new balance point = $l_2=64.8cm$

let the internal resistance of the cell be $r$.

Now as we know, in a potentiometer,

$r=\frac{l_1-l_2}{l_2}\ast R$

$r=\frac{76.3-64.8}{64.8}\ast 9.5=1.68\mathrm{\Omega }$

Hence, the internal resistance of the cell will be 1.68 $\mathrm{\Omega }$.

Chapter 3 of Class 12 Physics NCERT Solutions is crucial for students preparing for board exams, JEE, and NEET. This chapter introduces fundamental concepts like electric current and resistance, which are not only important for scoring well in board exams but also essential for competitive exams like JEE and NEET. The solutions help build a strong foundation for these topics, and understanding them is key for advancing in physics and electrical engineering. So, mastering this chapter is vital for both exam success and future studies!